Claim: the lateral (side ) surface of any area of any cylinder can is larger than the circlar surface area of its bottom. Which of the following is an example showing that the claim is false?
a. a can with the diameter of 5 inches and height of 10 inches
b. a can with diameter 1 inch and height 10 inches
c. a can with the diameter 10 inches and height of 2 inches
d. a can with diameter 10 inches and height of 10 inches.
The diameter must be much greater than the height.
C would be a false claim then ?
To determine if the claim is false, we need to compare the lateral surface area of any given cylinder with its circular surface area at the bottom.
The lateral surface area of a cylinder can be calculated using the formula: LSA = 2πrh, where r is the radius of the circular base and h is the height of the cylinder.
The circular surface area at the bottom of a cylinder can be calculated using the formula: CSA = πr^2, where r is the radius of the circular base.
Now, let's calculate the lateral surface area and circular surface area for each of the given examples:
a. a can with a diameter of 5 inches and height of 10 inches:
- Radius of the circular base (r) = 5/2 = 2.5 inches
- Lateral Surface Area (LSA) = 2πrh = 2π(2.5)(10) = 50π square inches
- Circular Surface Area (CSA) = πr^2 = π(2.5)^2 = 6.25π square inches
In this case, the LSA is larger than the CSA, thus disproving the claim. This example does not show that the claim is false.
b. a can with diameter 1 inch and height 10 inches:
- Radius of the circular base (r) = 1/2 = 0.5 inch
- Lateral Surface Area (LSA) = 2πrh = 2π(0.5)(10) = 10π square inches
- Circular Surface Area (CSA) = πr^2 = π(0.5)^2 = 0.25π square inches
In this case, the LSA is larger than the CSA, thus disproving the claim. This example does not show that the claim is false.
c. a can with diameter 10 inches and height of 2 inches:
- Radius of the circular base (r) = 10/2 = 5 inches
- Lateral Surface Area (LSA) = 2πrh = 2π(5)(2) = 20π square inches
- Circular Surface Area (CSA) = πr^2 = π(5)^2 = 25π square inches
In this case, the LSA is smaller than the CSA, thus disproving the claim. This example shows that the claim is false.
d. a can with diameter 10 inches and height of 10 inches:
- Radius of the circular base (r) = 10/2 = 5 inches
- Lateral Surface Area (LSA) = 2πrh = 2π(5)(10) = 100π square inches
- Circular Surface Area (CSA) = πr^2 = π(5)^2 = 25π square inches
In this case, the LSA is larger than the CSA, thus not disproving the claim. Therefore, this example does not show that the claim is false.
In conclusion, only example c, a can with a diameter of 10 inches and a height of 2 inches, disproves the claim that the lateral surface of any cylinder is larger than the circular surface area of its bottom.